Graph Theory: Favorite Conjectures and Open Problems - 2 PDF
Preview Graph Theory: Favorite Conjectures and Open Problems - 2
Problem Books in Mathematics Ralucca Gera · Teresa W. Haynes Stephen T. Hedetniemi Editors Graph Theory Favorite Conjectures and Open Problems - 2 Problem Books in Mathematics SeriesEditor: PeterWinkler DepartmentofMathematics DartmouthCollege Hanover,NH USA Moreinformationaboutthisseriesathttp://www.springer.com/series/714 Ralucca Gera • Teresa W. Haynes Stephen T. Hedetniemi Editors Graph Theory Favorite Conjectures and Open Problems - 2 123 Editors RaluccaGera TeresaW.Haynes DepartmentofAppliedMathematics DepartmentofMathematics NavalPostgraduateSchool EastTennesseeStateUniversity Monterey,CA,USA JohnsonCity,TN,USA StephenT.Hedetniemi SchoolofComputing ClemsonUniversity Clemson,SC,USA ISSN0941-3502 ISSN2197-8506 (electronic) ProblemBooksinMathematics ISBN978-3-319-97684-6 ISBN978-3-319-97686-0 (eBook) https://doi.org/10.1007/978-3-319-97686-0 LibraryofCongressControlNumber:2018959757 MathematicsSubjectClassification(2010):05Cxx,01-02,01-08,03Fxx ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents Introduction ...................................................................... 1 RaluccaGera,StephenT.Hedetniemi,andTeresaW.Haynes DesertIslandConjectures....................................................... 7 LowellW.Beineke BindingNumber,Cycles,andCliques ......................................... 19 WayneGoddard AConjectureonLaplacianEigenvaluesofTrees............................. 27 DavidP.JacobsandVilmarTrevisan QueensAroundtheWorldinTwenty-FiveYears............................. 43 WilliamD.Weakley ReflectionsonaThemeofUlam................................................ 55 RonGraham UlamNumbersofGraphs....................................................... 63 StephenT.Hedetniemi ForbiddenTrees.................................................................. 69 DavidSumner SomeofMyFavouriteConjectures:LocalConditionsImplying GlobalCycleProperties......................................................... 91 OrtrudR.Oellermann ThePathPartitionConjecture.................................................. 101 MarietjieFrickandJeanE.Dunbar TotheMoonandBeyond........................................................ 115 EllenGethner MyFavoriteDominationGameConjectures.................................. 135 MichaelA.Henning v vi Contents ADeBruijn–Erdo˝sTheoreminGraphs?..................................... 149 VašekChvátal AnAnnotatedGlossaryofGraphTheoryParameters, withConjectures ................................................................. 177 Ralucca Gera, Teresa W. Haynes, Stephen T. Hedetniemi, andMichaelA.Henning Introduction RaluccaGera,StephenT.Hedetniemi,andTeresaW.Haynes 1 Conjectures andOpenProblems Thisbookisthesecondinatwo-volumeseriesonconjecturesandopenproblemsin graphtheory.Theprimarymotivation,themeandvisionoftheserieswasexpressed in the Introductionto Volume I, a slightly revised version of which we reproduce here. The series has its roots in the idea that conjecturesare central to mathematics, and that it is useful to periodically identify and survey conjectures in the various branches of mathematics. Typically, the end results of mathematics research are theorems,themostimportantandfamousofwhichshowupintextbooks,whichin turnare taughtto students. Thisoftengivesstudentsthe impressionthattheorems arethemostimportantthingsinmathematics.Thepopularpressreinforcesthisidea; when mathematicsis in the newspapersit is most often to reporta proof of some well-known,unsolvedconjectureorproblem. However, as every research mathematician knows, progress in mathematics involvesmuchmorethanprovingtheorems,anditspracticeismuchricher.Math- ematics research involves not only proving theorems, but also raising questions, formulating open problems, and stating the conjectures, the solutions to which become the new theorems. Mathematics research also involves the formation of new conceptsand methods,the productionof counterexamplesto conjectures,the R.Gera((cid:2)) DepartmentofAppliedMathematics,NavalPostgraduateSchool,Monterey,CA,USA e-mail:[email protected] S.T.Hedetniemi SchoolofComputing,ClemsonUniversity,Clemson,SC,USA T.W.Haynes DepartmentofMathematics,EastTennesseeStateUniversity,JohnsonCity,TN,USA ©SpringerNatureSwitzerlandAG2018 1 R.Geraetal.(eds.),GraphTheory,ProblemBooksinMathematics, https://doi.org/10.1007/978-3-319-97686-0_1 2 R.Geraetal. simplificationandsynthesisofdifferentareasofmathematics,andthedevelopment ofanalogiesacrossdifferentareasofmathematics. Thethreeeditorsofthisvolumehappentobegraphtheorists,ormoregenerally discretemathematicians,whoexplainthemajorfocusofthefollowingchapters.In this collection of papers, the contributing authors present and discuss, often in a story-telling style, some of the most well-known conjectures in the field of graph theoryandcombinatorics. Related to conjectures are open problems. Conjectures are either true or false. Butwhatcountsastheresolutionofaproblemisoftenlessclear-cut.Nevertheless, a conjecture clearly specifies a problem—and many problems can be naturally formulated as conjectures. For example, it is a famous unsolved problem to determinewhetherornottheclassPofdecisionproblemsisequaltoitssuperclass NP.ThefamousP=NPproblemisoneofthesevenMillenniumProblemsidentified bytheClayMathematicsInstitute,whoseresolutioncarriesa$1milliondollarprize (http://www.claymath.org/millennium-problems/rules-millennium-prizes). For any problemlikethis,havingayesornooutcome,theassociatedconjecturesareeither thattheproblemcanberesolvedinthepositiveorthatitcannotbe.Manyormost mathematicians, for instance, conjecture that P(cid:2)=NP. But a few, including Bela Bollabás,conjecturethatP=NP. While most mathematicians are most likely to be known for their theorems, someareknownfortheirconjectures.FermatandPoincaré,whilefamousfortheir theorems,are also knownfor their conjectures.Graphtheoristsknowthe nameof Francis Guthrie only for his conjecturethat planar maps can be coloredwith four colors[10]. Theworld-famousmathematicianPaulErdösis anexceptionalexample.While he is known for, among other things, his development of Ramsey theory, the probabilistic method, and contributions to the elementary proof of the prime number theorem, he is perhaps equally famous for his conjectures and problems. He travelled with these, talked about them, worked on them with hundreds of collaborators,andevenofferedmonetaryprizesforthesolutionsofmanyofthem. Someofhisgraphtheoryconjecturesarecollectedin[3].Hisconjecturesandprizes have inspired considerable research and numerousresearch papers, and still more than20yearsafterhisdeathin1996hisconjecturescontinuetohaveconsiderable influence. Notallconjecturesareofequalimportanceorsignificance,andnotallwillhave the same influence on mathematics research. The resolution of some conjectures willimpacttextbooksandeventhehistoryofmathematics.Theresolutionofothers will soon be forgotten. What makes a conjecture significant or important? A few mathematicianshaverecordedtheirthoughtsonthisquestion. The famous British mathematician, G.H. Hardy, the early twentieth century analyst and number theorist, discussed this question in his 1940 essay, A Mathe- matician’sApology[8],whichisabiographicaldefenseofmathematicsashesaw andpracticedit.Hardyisoftenrememberedfordiscountingthepracticalityorutility ofmathematics. Introduction 3 LaszloLovászhasdiscussedthequestionofwhatmakesagoodconjecture[9]. He says that “it is easy to agree that” the resolutionof a good conjecture“should advance our knowledge significantly.” Nevertheless, Lovász wants to make room for some of the conjecturesof Erdös that don’tobviously satisfy this criteria, but are“conjecturessosurprising,soutterlyinaccessiblebycurrentmethods,thattheir resolutionmustbringsomethingnew—wejustdon’tknowwhere.” Lovász also discusses experimental mathematics as a source of conjectures, a specificexampleofwhichbeingFajtlowicz’sGraffiti[4],acomputerprogramwhich makesconjectures,manyofwhichareingraphtheory.Itiseasytowriteaprogram toproducesyntacticallycorrectmathematicalstatements.Thedifficultyinwritinga mathematical conjecture-making program is exactly how to limit the program to making interesting or significant statements. When Fajtlowicz began writing his program he would ask mathematicians what constituted a good conjecture. John Conway told him that a good conjectureshould be “outrageous.”Erdös, in effect, refusedtoanswer,tellingFajtlowicz,“Let’sleaveittoRadymanthus.” We won’t here give a definitive answer to the question: what makes a good conjecture?Fame isneithera necessarynora sufficientconditionfor a conjecture to be considered good. Sociology plays some role in fame. The nonexistence of oddperfectnumbersisprobablymorefamousduetoitsage,datingbacktoEuclid and later to Descartes, than its importance [11]. But many conjectures of famous mathematiciansareworkedonbecauseoftheirintrinsicimportancetomathematics. We certainly expect there to exist little known but significant conjectures. The history of mathematics contains numerous examples of important research which was not recognized in its own time. The work of Galois, for instance, is a well- knownexample. Our thought is that there may not be any better way of identifying good conjectures than to ask the experts, people who have tilled the mathematical soil forsometime,andknowbestwhichseedswillsprout. There are also internal reasons for the importance of a conjecture, strictly mathematicalreasons related to the furtheranceof mathematicalresearch, and the question,howwouldresolutionofagivenconjectureadvancemathematics? Ofcoursethegoalofmathematicalresearch,andwhatresearchmathematicians are paidto do,is to advancemathematics.Mathematicsis seen by the publicas a toolforthesciences—theywouldhavemuchlessinterestinpayingmathematicians tobeartiststhantheywouldasresearcherswhomayplayaroleinimprovingtheir lives. Buthowcanaconjectureplayaroleinadvancingmathematics?Inparticularit mayseemthatwehaveanewquestiontoaddress:howdoesmathematicsadvance? A conjecture can be said to advance mathematics if the truth of the conjecture yieldsnewknowledgeaboutaquestionorobjectofexistingmathematicalinterest. Furthermore, the advancement of mathematics requires not just new concepts, conjectures,counterexamples,andproofs(uniquelymathematicalproducts)butalso effectivecommunication.Lovászwrites: Conjecture-makingisoneof thecentralactivitiesin mathematics.Thecreation and dissemination of open problems is crucial to the growth and developmentof
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